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The organization of the present paper is as follows. In the section 2, we recall the definition and determinant of the square Cauchy’s and Hilbert’s matrix over a field of characteristic zero. In section 3, we calculate the determinant of a certain square matrix that are related to the Cauchy’s matrix. Finally, in Section 4, we use the result of Section 3 to prove Theorem 1.1.
2. Cauchy’s and Toeplitz matrices In 1841, Augustin Louis Cauchy introduced a certain type of matrices with certain properties, see [4, 5]. We are going to recall the definition and determinant of these matrices in this section. An n × n square Cauchy’s matrix defined by disjoint subsets of distinct nonzero elements {x1, · · · ,xn} and {y1, · · · ,yn} in a field of characteristic zero F , is the square matrix Xn := [xij] with 1 xij = , 1 ≤ i, j ≤ n. xi − yj Note that any submatrix of a Cauchy’s matrix is itself a Cauchy’s matrix. The determinant of a Cauchy’s matrix is known as Cauchy’s determinant in the literature, which is always nonzero because xi 6= yj. Following proposi- tion shows that how one can calculate the determinant of Cauchy’s matrices.
Proposition 2.1. Let n ≥ 1 be an integer and Xn a n × n Cauchy’s matrix defined as above over a field F of characteristic zero. Then
(xi − xj)(yi − yj) |X | = i ′ ′ (x1 − xi) 1 xi1 = 0, xij = · 2 ≤ i, j ≤ n. (x1 − yj) (xi − yj) ON SPECIAL MATRICES RELATED TO CAUCHY AND TOEPLITZ MATRICES 3 Extracting the factor (x1 − xi) from each rows, and 1/(x1 − yj) from each column, for 2 ≤ i, j ≤ n, gives that 1 1 · · · 1 (x2−y2) (x2−y3) (x2−yn) n 1 1 · · · 1 1 (y1 − yj)(x1 − xi) (x3−y2) (x3−y3) (x3−yn) |X | = . n (x − y ) (x − y )(x − y ) . . . 1 1 i,jY=2 i 1 1 i . · · · . . 1 1 1 · · · (xn−y2) (xn−y3) (xn−yn) Repeating this procedure, we obtain that 1 (yi − yj)(xj − xi) (xi − xj)(yi − yj) |X | = · i In [1], Hilbert introduced a certain square matrix which is a special case of the Cauchy square matrix. The Hilbert’s matrix is an n×n matrix Hn = [hij] with entries hij = 1/(i + j − 1), where 1 ≤ i, j ≤ n. Using the proposition 2.1, one can calculate the determinant of a Hilbert’s matrix as n−1 c4 |H | = n , c = i!. n c n 2n Yi=1 He also mentioned that the determinant of Hn is the reciprocal of a well known integer which follows from the following identity 2n−1 1 c i = 2n = n! · . |H | c4 [i/2] n n Yi For more information see the sequence A005249 in OEIS [8]. For a recent work related to the Cauchy’s and Hilbert’s matrices one can see [10]. The other type of matrices, which we are going to recall here, are the Toeplitz matrices. An n × n Toeplitz matrix with entries in a field F is the square matrix v0 v1 v2 · · · vn−1 v−1 v0 v1 · · · vn−2 v v v · · · v Vn := −2 −1 0 n−3 . . . . . . . . · · · . v1−n v2−n v3−n · · · v0 These are one of the most well studied and understood classes of ma- trices that arise in most areas of the mathematics: algebra [11], algebraic geometry [12], and graph theory [13]. In [3], the author obtained a unique LU factorizations and an explicit formula for the determinant and also the inversion of Toeplitz matrices. And, the inverse, determinants, eigenvalues, and eigenvectors of symmetric Toeplitz matrices over real number field with linearly increasing entries have been studied in [14]. In [15], the author 4 SAJAD SALAMI showed that every n × n square matrix is generically a product of ⌊n/2⌋ + 1 and always a product of at most 2n + 5 Toeplitz matrices. 3. Determinant of certain square matrix In this section, we calculate the determinant of certain square matrices with entries in a field F of characteristic zero, which are related to the determinant of Cauchy’s matrix. In special case, the determinant of our matrix is related to the determinant of a certain Toeplitz matrix. First, let us to give the following elementary result for a given infinite sequence ∞ {aℓ}ℓ=1 of distinct nonzero elements in a field F of characteristic zero. Lemma 3.1. For indexes e, ℓ, s, and t, we have d(t,e) d(s,e) d(t,s)d(ℓ,e) asd(ℓ,e) − aℓd(s,e) = −aed(s,ℓ), − = . d(t,ℓ) d(s,ℓ) d(t,ℓ)d(s,ℓ) Proof. For indexes e, ℓ, and s, by definition d(s,e) = d(s,ℓ) + d(ℓ,e), so asd(ℓ,e) − aℓd(s,e) = asd(ℓ,e) − aℓ(d(s,ℓ) + d(ℓ,e)) = (as − aℓ)d(ℓ,e) − aℓd(s,ℓ) = d(s,ℓ)(d(ℓ,e) − aℓ)= −aed(s,ℓ) For indexes e, ℓ, s, and t, one has d(t,e) d(s,e) d(t,e)d(s,ℓ) − d(s,e)d(t,ℓ) − = d(t,ℓ) d(s,ℓ) d(t,ℓ)d(s,ℓ) 1 d d = · (t,t) (t,ℓ) d d d d (t,ℓ) (s,ℓ) (s,e) (s,ℓ) 1 d − d d − d = · (t,e) (s,e) (t,ℓ) (s,ℓ) d d d d (t,ℓ) (s,ℓ) (s,e) (s,ℓ) 1 d d = · (t,s) (t,s) d d d d (t,ℓ) (s,ℓ) (s,e) (s,ℓ) d(t,s) 1 1 = · d d d d (t,ℓ) (s,ℓ) (s,e) (s,ℓ) d(t,s)(d(s,ℓ) − d(s,e)) d( t,s)d(ℓ,e) = = . d(t,ℓ)d(s,ℓ) d(t,ℓ)d(s,ℓ) ON SPECIAL MATRICES RELATED TO CAUCHY AND TOEPLITZ MATRICES 5 For any integer n ≥ 1, define (n + 1) × (n + 1) matrix An as: a a a 1 i1 i1 · · · i1 d(i ,e ) d(i ,e ) d(i ,en) a1 1 a1 2 a1 1 i2 i2 · · · i2 d(i2,e1) d(i2,e2) d(i2,en) . . . . An := . . · · · . . , ain ain ain 1 · · · d(in,e ) d(in,e ) d(in,en) a 1 a 2 a in+1 in+1 in+1 1 d d · · · d (in+1,e1) (in+1,e2) (in+1,en) where {ai1 , · · · , ain+1 } and {ae1 , · · · , aen } are disjoint subsets of the infinite ∞ sequence {aℓ}ℓ=1. The following proposition gives the determinant of An. We will use Lemma 3.1 in its proof. Proposition 3.2. Let I = {1, 2, · · · ,n} and J = {1, 2, · · · ,n + 1}. Then, one has Dr · ′ d(i ,i ′ ) |A | = s By extracting the factor −aej /d(i1,ej ) from each columns (1 ≤ j ≤ n) and d(is,i1) from each rows (2 ≤ s ≤ n + 1), one gets that n n+1 ae |A | = (−1)n j · d · |B | n d (is,i1) n jY=1 (i1,ej ) sY=2 where 1 1 · · · 1 d(i2,e1) d(i2,e2) d(i2,en) 1 1 1 d d · · · d B := (i3,e1) (i3,e2) (i3,en) n . . . . · · · . . 1 1 1 d d · · · d . (in+1,e1) (in+1,e2) (in+1,en) Since the matrix Bn is a Cauchy’s matrix defined by x1 = ai2 , · · · ,xn = ain+1 , y1 = ae1 , · · · ,yn = aen , 6 SAJAD SALAMI so using Proposition 2.1 we have ′ s n n aej (−1) · ′ d(i ,i ′ ) j=1 d(i ,e ) s We note that the matrix Bn in the proof of the above proposition is related to a certain n × n Toeplitz matrix. Indeed, if we consider the sequence aℓ = 1/ℓ for ℓ = 1, 2, · · · and indexes ej = j and is = n+s−1 for j = 1, · · · ,n and n s = 1, · · · ,n + 1, then a simple calculation shows that Bn = (−1) (2n)!Vn, where Vn is the following n × n Toeplitz matrix 1 1 1 1 n n−1 n−2 · · · 2 1 1 1 1 1 1 n+1 n n−1 · · · 3 2 1 1 1 · · · 1 1 n+2 n+1 n 4 3 k Vn = . . . . . = (−1) Hn, . . . · · · . . 1 1 1 · · · 1 1 2n−2 2n−3 2n−4 n n−1 1 1 1 · · · 1 1 2n−1 2n−2 2n−3 n+1 n where k = n/2 if n is even and k = (n − 1)/2 if n is odd; and the last equality comes by changing j-th column with (n − j + 1)-th column of Vn. 4. Proof of theorem 1.1 In order to prove Theorem 1.1, we need the following result. Proposition 4.1. Given integers 2 ≤ r 2 |C′| = (−1)r +3rDr+1 d . r (is,is′ ) s′ ′ 3r(r+1)/2 ′′ |C | = (−1) d(ℓ,e) aℓd(ℓ,e) aℓd(ℓ,e) · |C | e<ℓY∈I e<ℓY∈I1 e<ℓY∈Ir r(r+3)/2 r ′′ = (−1) Dr · |C |, ON SPECIAL MATRICES RELATED TO CAUCHY AND TOEPLITZ MATRICES 7 where C′′ is the following (r + 1) × (r + 1) matrix d a d · · · a d ℓ∈I (i1,ℓ) i1 ℓ∈I1 (i1,ℓ) i1 ℓ∈Ir (i1,ℓ) d a d · · · a d ′′ Qℓ∈I (i2,ℓ) i2 Qℓ∈I1 (i2,ℓ) i2 Qℓ∈Ir (i2,ℓ) C := Q . Q . Q . . . · · · . . d ai d · · · ai d ℓ∈I (ir+1,ℓ) r+1 ℓ∈I1 (ir+1,ℓ) r+1 ℓ∈Ir (ir+1,ℓ) Q Q Q By extracting the factor ℓ∈I d(is,ℓ) from s-th row 1 ≤ s ≤ r + 1, we obtain Q ai ai 1 1 · · · 1 d(i1,1) d(i1,r) ai2 ai2 1 d · · · d |C′′| = d · (i2,1) (i2,r) (is,j) . . . sY∈J Yj∈I . . · · · . a a 1 ir+1 · · · ir+1 d(i ,1) d(i ,r) r+1 r+1 Considering t = r and ej = j for j = 1, · · · , r, and using Proposition 3.2 for calculating the last determinant, one can conclude that Dr · ′ d(i ,i ′ ) |C′| = (−1)r(r+3)/2Dr−1 d · s We notice that above proposition is a special case of the next general one. Proposition 4.2. Given integers 2 ≤ r C(i1−r,0) C(i1−r,j2) · · · C(i1−r,jm) C(i2−r,0) C(i2−r,j2) · · · C(i2−r,jm) Cm = . . . , . . · · · . C C · · · C . (im−r,0) (im−r,j2) (im−r,jm) such that r C(is−r,0) = (−1) d(is,ℓ)d(ℓ,e), e<ℓY∈I C = (−1)r+js′ a a d d , (is−r,js′ ) is ℓ (is,ℓ) (ℓ,e) e<ℓY∈Ij s′ 8 SAJAD SALAMI ′ r where I = {1, 2, · · · , r} and Ijs′ = I\{js l}. Extracting (−1) e<ℓ∈I d(ℓ,e) r+js′ ′ and (−1) e<ℓ∈Ij aℓd(ℓ,e) from first and s -th columns,Q respectively, s′ Q ′ and then ℓ∈I d(is,ℓ) from s-th row for 1 ≤ s e<ℓY∈I Yt=2 e<ℓY∈Ijt sY=1 Yℓ∈I a a 1 i1 · · · i1 d(i ,j ) d(i ,jm) a1 2 a1 1 i2 · · · i2 d(i ,j ) d(i ,j ) × 2 2 2 m , . . . . . · · · . aim aim 1 d · · · d (im,j2) (im,jm) ′ where r = mr + j2 + · · · + jm and the above is nonzer by Propositions 3.2. Otherwise, if suppose that 1 ≤ j1 < j2 < · · · < jm, then extracting the r+js′ ′ factor (−1) ais e<ℓ∈I aℓd(ℓ,e) from s -th column, and then ℓ∈I d(is,ℓ) js′ Q ′ Q from s-th row of the matrix Cm = [Cis−r,js′ ], where 1 ≤ s,s ≤ m, gives that m r′′ |Cm| = (−1) aℓd(ℓ,e)d(is,ℓ) s,tY=1 e<ℓY∈Ijt 1 1 · · · 1 d(i1,j1) d(i1,j2) d(i1,jm) 1 1 · · · 1 d(i ,j ) d(i ,j ) d(i ,jm) × 2 1 2 2 2 , . . . . · · · . . 1 1 1 d d · · · d (im,j1) (im,j2) (im,jm) ′′ where r = mr + j1 + · · · + jm and the last determinant is nonzero by Propositions 2.1. This completes the proof of the proposition. Now we are ready to prove the main theorem 1.1, using the above results. n Proof. For integers 2 ≤ r References [1] Hilbert D. Ein betrag zur theoreie des Legendre’schen polynoms. Acta Mathematica, Vol. 18, 155-159, (1894). ON SPECIAL MATRICES RELATED TO CAUCHY AND TOEPLITZ MATRICES 9 [2] Choi M-D. Tricks or Treats with the Hilbert Matrix. Amer. Math. Month., Vol. 90, No. 5, 301-312, 1983. [3] Li HSUAN-CHU On Calculating the Determinants of Toeplitz Matrices. Journal of Applied Mathematics and Bioinformatics, Vol. 1, No. 1, 55-64 (2011). [4] Cauchy AL. M´emorie sur les fonctions altern´ees et sur les somme altern´ees. Exercises d Analyse et de Phys. Math., Vol. II, 151-159, (1841). [5] P´olya G, Szego G. Zweiter Band. Springer, Berlin, Vol., (1925). [6] Salami S. Rational points on a certain family of complete intersection varieties. Under Preparation (2019). [7] Lang S. Number Theory III: Survey of Diophantine Geometry. Encyclopaedia of Mathematical Sciences, Springer, Berlin, Vol. 60, (1991). [8] Sloane N.J.A. The On-Line Encyclopedia of Integer Sequences. http://oeis.org. Se- quence A005249. [9] Davis PH.J. Interpolation and approximation. Dover Publication Inc., New-Yourk (NY) (1975). [10] Fiedle M. Notes on Hilbert and Cauchy matrices, Linear Algebra and its Applications, Vol. 432, 351-356, (2010). [11] Rietsch K. Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Amer. Math. Soc., Vol. 16, no. 2. 2003. p. 363-392. [12] Englis M. Toeplitz operators and group representations. J. Fourier Anal. Appl., Vol. 13, no. 3, 243-265, (2007). [13] Euler R. Characterizing bipartite Toeplitz graphs. Theoret. Comput. Sci., Vol. 263, no. 1-2, 47-58, (2001). [14] Bunger F. Inverse, determinants, eigenvalues, and eigenvectors of real symmetric Toeplitz matrices with linearly increasing entrie. Linear Algebra and its Applications, Vol. 459, 595-619, (2014). [15] Ye KE, Lim LH. Every Matrix is a Product of Toeplitz Matrices. Found. Comput. Math., Vol. 16, no. 1-2, 577-598, (2016). (Sajad Salami) Inst´ıtuto da Matematica´ e Estat´ıstica, Universidade Estadual do Rio do Janeiro, Brazil E-mail address, Sajad Salami: [email protected] URL: https://sites.google.com/a/ime.uerj.br/sajadsalami/